Open Access

Power series method and approximate linear differential equations of second order

Advances in Difference Equations20132013:76

DOI: 10.1186/1687-1847-2013-76

Received: 14 December 2012

Accepted: 4 March 2013

Published: 26 March 2013

Abstract

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

MSC:34A05, 39B82, 26D10, 34A40.

Keywords

power series method approximate linear differential equation simple harmonic oscillator equation Hyers-Ulam stability approximation

1 Introduction

Let X be a normed space over a scalar field K , and let I R be an open interval, where K denotes either or . Assume that a 0 , a 1 , , a n : I K and g : I X are given continuous functions. If for every n times continuously differentiable function y : I X satisfying the inequality
a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) ε
for all x I and for a given ε > 0 , there exists an n times continuously differentiable solution y 0 : I X of the differential equation
a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) = 0

such that y ( x ) y 0 ( x ) K ( ε ) for any x I , where K ( ε ) is an expression of ε with lim ε 0 K ( ε ) = 0 , then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [18].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9, 10]). Thereafter, Alsina and Ger [11] proved the Hyers-Ulam stability of the differential equation y ( x ) = y ( x ) . It was further proved by Takahasi et al. that the Hyers-Ulam stability holds for the Banach space valued differential equation y ( x ) = λ y ( x ) (see [12] and also [1315]).

Moreover, Miura et al. [16] investigated the Hyers-Ulam stability of an n th-order linear differential equation. The first author also proved the Hyers-Ulam stability of various linear differential equations of first order (ref. [1725]).

Recently, the first author applied the power series method to studying the Hyers-Ulam stability of several types of linear differential equations of second order (see [2634]). However, it was inconvenient that he had to alter and apply the power series method with respect to each differential equation in order to study the Hyers-Ulam stability. Thus, it is inevitable to develop a power series method that can be comprehensively applied to different types of differential equations.

In Sections 2 and 3 of this paper, we establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

Throughout this paper, we assume that the linear differential equation of second order of the form
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = 0 ,
(1)
for which x = 0 is an ordinary point, has the general solution y h : ( ρ 0 , ρ 0 ) C , where ρ 0 is a constant with 0 < ρ 0 and the coefficients p , q , r : ( ρ 0 , ρ 0 ) C are analytic at 0 and have power series expansions
p ( x ) = m = 0 p m x m , q ( x ) = m = 0 q m x m and r ( x ) = m = 0 r m x m

for all x ( ρ 0 , ρ 0 ) . Since x = 0 is an ordinary point of (1), we remark that p 0 0 .

2 Inhomogeneous differential equation

In the following theorem, we solve the linear inhomogeneous differential equation of second order of the form
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m
(2)

under the assumption that x = 0 is an ordinary point of the associated homogeneous linear differential equation (1).

Theorem 2.1 Assume that the radius of convergence of power series m = 0 a m x m is ρ 1 > 0 and that there exists a sequence { c m } satisfying the recurrence relation
k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m
(3)
for any m N 0 . Let ρ 2 be the radius of convergence of power series m = 0 c m x m and let ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } , where ( ρ 0 , ρ 0 ) is the domain of the general solution to (1). Then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the linear homogeneous differential equation (1).

Proof Since x = 0 is an ordinary point, we can substitute m = 0 c m x m for y ( x ) in (2) and use the formal multiplication of power series and consider (3) to get
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 k = 0 m p m k ( k + 2 ) ( k + 1 ) c k + 2 x m + m = 0 k = 0 m q m k ( k + 1 ) c k + 1 x m + m = 0 k = 0 m r m k c k x m = m = 0 k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] x m = m = 0 a m x m
for all x ( ρ 3 , ρ 3 ) . That is, m = 0 c m x m is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution y : ( ρ 3 , ρ 3 ) C of (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m ,

where y h ( x ) is a solution of the linear homogeneous differential equation (1). □

For the most common case in applications, the coefficient functions p ( x ) , q ( x ) , and r ( x ) of the linear differential equation (1) are simple polynomials. In such a case, we have the following corollary.

Corollary 2.2 Let p ( x ) , q ( x ) , and r ( x ) be polynomials of degree at most d 0 . In particular, let d 0 be the degree of p ( x ) . Assume that the radius of convergence of power series m = 0 a m x m is ρ 1 > 0 and that there exists a sequence { c m } satisfying the recurrence formula
k = m 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m
(4)
for any m N 0 , where m 0 = max { 0 , m d } . If the sequence { c m } satisfies the following conditions:
  1. (i)

    lim m c m 1 / m c m = 0 ,

     
  2. (ii)

    there exists a complex number L such that lim m c m / c m 1 = L and p d 0 + L p d 0 1 + + L d 0 1 p 1 + L d 0 p 0 0 ,

     
then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where ρ 3 = min { ρ 0 , ρ 1 } and y h ( x ) is a solution of the linear homogeneous differential equation (1).

Proof Let m be any sufficiently large integer. Since p d + 1 = p d + 2 = = 0 , q d + 1 = q d + 2 = = 0 and r d + 1 = r d + 2 = = 0 , if we substitute m d + k for k in (4), then we have
a m = k = 0 d [ ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k + ( m d + k + 1 ) c m d + k + 1 q d k + c m d + k r d k ] .
By (i) and (ii), we have
lim sup m | a m | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 × ( p d k + q d k ( m d + k + 2 ) c m d + k + 1 c m d + k + 2 + r d k ( m d + k + 2 ) ( m d + k + 1 ) c m d + k c m d + k + 1 c m d + k + 1 c m d + k + 2 ) | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | k = d d 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | ( m d 0 + 2 ) ( m d 0 + 1 ) c m d 0 + 2 ( p d 0 + L p d 0 1 + + L d 0 p 0 ) | 1 / m = lim sup m | ( p d 0 + L p d 0 1 + + L d 0 p 0 ) ( m d 0 + 2 ) ( m d 0 + 1 ) | 1 / m × ( | c m d 0 + 2 | 1 / ( m d 0 + 2 ) ) ( m d 0 + 2 ) / m = lim sup m | c m d 0 + 2 | 1 / ( m d 0 + 2 ) ,

which implies that the radius of convergence of the power series m = 0 c m x m is ρ 1 . The rest of this corollary immediately follows from Theorem 2.1. □

In many cases, it occurs that p ( x ) 1 in (1). For this case, we obtain the following corollary.

Corollary 2.3 Let ρ 3 be a distance between the origin 0 and the closest one among singular points of q ( z ) , r ( z ) , or m = 0 a m z m in a complex variable z. If there exists a sequence { c m } satisfying the recurrence relation
( m + 2 ) ( m + 1 ) c m + 2 + k = 0 m [ ( k + 1 ) c k + 1 q m k + c k r m k ] = a m
(5)
for any m N 0 , then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation
y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m
(6)
can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the linear homogeneous differential equation (1) with p ( x ) 1 .

Proof If we put p 0 = 1 and p i = 0 for each i N , then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that m = 0 c m x m is a particular solution of the linear inhomogeneous differential equation (6).

According to [[35], Theorem 7.4] or [[36], Theorem 5.2.1], there is a particular solution y 0 ( x ) of (6) in a form of power series in x whose radius of convergence is at least ρ 3 . Moreover, since m = 0 c m x m is a solution of (6), it can be expressed as a sum of both y 0 ( x ) and a solution of the homogeneous equation (1) with p ( x ) 1 . Hence, the radius of convergence of m = 0 c m x m is at least ρ 3 .

Now, every solution y : ( ρ 3 , ρ 3 ) C of (6) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m ,

where y h ( x ) is a solution of the linear differential equation (1) with p ( x ) 1 . □

3 Approximate differential equation

In this section, let ρ 1 > 0 be a constant. We denote by C the set of all functions y : ( ρ 1 , ρ 1 ) C with the following properties:
  1. (a)

    y ( x ) is expressible by a power series m = 0 b m x m whose radius of convergence is at least ρ 1 ;

     
  2. (b)
    There exists a constant K 0 such that m = 0 | a m x m | K | m = 0 a m x m | for any x ( ρ 1 , ρ 1 ) , where
    a m = k = 0 m [ ( k + 2 ) ( k + 1 ) b k + 2 p m k + ( k + 1 ) b k + 1 q m k + b k r m k ]
     

for all m N 0 and p 0 0 .

Lemma 3.1 Given a sequence { a m } , let { c m } be a sequence satisfying the recurrence formula (3) for all m N 0 . If p 0 0 and n 2 , then c n is a linear combination of a 0 , a 1 , , a n 2 , c 0 , and c 1 .

Proof We apply induction on n. Since p 0 0 , if we set m = 0 in (3), then
c 2 = 1 2 p 0 a 0 r 0 2 p 0 c 0 q 0 2 p 0 c 1 ,
i.e., c 2 is a linear combination of a 0 , c 0 , and c 1 . Assume now that n is an integer not less than 2 and c i is a linear combination of a 0 , , a i 2 , c 0 , c 1 for all i { 2 , 3 , , n } , namely,
c i = α i 0 a 0 + α i 1 a 1 + + α i i 2 a i 2 + β i c 0 + γ i c 1 ,
where α i 0 , , α i i 2 , β i , γ i are complex numbers. If we replace m in (3) with n 1 , then
a n 1 = 2 c 2 p n 1 + c 1 q n 1 + c 0 r n 1 + 6 c 3 p n 2 + 2 c 2 q n 2 + c 1 r n 2 + + n ( n 1 ) c n p 1 + ( n 1 ) c n 1 q 1 + c n 2 r 1 + ( n + 1 ) n c n + 1 p 0 + n c n q 0 + c n 1 r 0 = ( n + 1 ) n p 0 c n + 1 + [ n ( n 1 ) p 1 + n q 0 ] c n + + ( 2 p n 1 + 2 q n 2 + r n 3 ) c 2 + ( q n 1 + r n 2 ) c 1 + r n 1 c 0 ,
which implies
c n + 1 = 1 ( n + 1 ) n p 0 a n 1 n ( n 1 ) p 1 + n q 0 ( n + 1 ) n p 0 c n 2 p n 1 + 2 q n 2 + r n 3 ( n + 1 ) n p 0 c 2 q n 1 + r n 2 ( n + 1 ) n p 0 c 1 r n 1 ( n + 1 ) n p 0 c 0 = α n + 1 0 a 0 + α n + 1 1 a 1 + + α n + 1 n 1 a n 1 + β n + 1 c 0 + γ n + 1 c 1 ,

where α n + 1 0 , , α n + 1 n 1 , β n + 1 , γ n + 1 are complex numbers. That is, c n + 1 is a linear combination of a 0 , a 1 , , a n 1 , c 0 , c 1 , which ends the proof. □

In the following theorem, we investigate a kind of Hyers-Ulam stability of the linear differential equation (1). In other words, we answer the question whether there exists an exact solution near every approximate solution of (1). Since x = 0 is an ordinary point of (1), we remark that p 0 0 .

Theorem 3.2 Let { c m } be a sequence of complex numbers satisfying the recurrence relation (3) for all m N 0 , where (b) is referred for the value of a m , and let ρ 2 be the radius of convergence of the power series m = 0 c m x m . Define ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } , where ( ρ 0 , ρ 0 ) is the domain of the general solution to (1). Assume that y : ( ρ 1 , ρ 1 ) C is an arbitrary function belonging to C and satisfying the differential inequality
| p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) | ε
(7)
for all x ( ρ 3 , ρ 3 ) and for some ε > 0 . Let α n 0 , α n 1 , , α n n 2 , β n , γ n be the complex numbers satisfying
c n = α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 + β n c 0 + γ n c 1
(8)
for any integer n 2 . If there exists a constant C > 0 such that
| α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 | C | a n |
(9)
for all integers n 2 , then there exists a solution y h : ( ρ 3 , ρ 3 ) C of the linear homogeneous differential equation (1) such that
| y ( x ) y h ( x ) | C K ε

for all x ( ρ 3 , ρ 3 ) , where K is the constant determined in (b).

Proof By the same argument presented in the proof of Theorem 2.1 with m = 0 b m x m instead of m = 0 c m x m , we have
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m
(10)
for all x ( ρ 3 , ρ 3 ) . In view of (b), there exists a constant K 0 such that
m = 0 | a m x m | K | m = 0 a m x m |
(11)

for all x ( ρ 1 , ρ 1 ) .

Moreover, by using (7), (10), and (11), we get
m = 0 | a m x m | K | m = 0 a m x m | K ε

for any x ( ρ 3 , ρ 3 ) . (That is, the radius of convergence of power series m = 0 a m x m is at least ρ 3 .)

According to Theorem 2.1 and (10), y ( x ) can be written as
y ( x ) = y h ( x ) + n = 0 c n x n
(12)

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the c n can be expressed by a linear combination of the form (8) for each integer n 2 .

Since n = 0 c n x n is a particular solution of (2), if we set c 0 = c 1 = 0 , then it follows from (8), (9), and (12) that
| y ( x ) y h ( x ) | n = 0 | c n x n | C K ε

for all x ( ρ 3 , ρ 3 ) . □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

Authors’ Affiliations

(1)
Mathematics Section, College of Science and Technology, Hongik University
(2)
Department of Mathematics, Faculty of Sciences and Arts, Istanbul Commerce University

References

  1. Brillouet-Belluot N, Brzdȩk J, Cieplinski K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012., 2012: Article ID 716936
  2. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge; 2002.View Article
  3. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView Article
  4. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston; 1998.View Article
  5. Hyers DH, Rassias TM: Approximate homomorphisms. Aequ. Math. 1992, 44: 125–153. 10.1007/BF01830975MathSciNetView Article
  6. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.View Article
  7. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View Article
  8. Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.
  9. Obłoza M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13: 259–270.
  10. Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 1997, 14: 141–146.
  11. Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2: 373–380.MathSciNet
  12. Takahasi S-E, Miura T, Miyajima S:On the Hyers-Ulam stability of the Banach space-valued differential equation y = λ y . Bull. Korean Math. Soc. 2002, 39: 309–315. 10.4134/BKMS.2002.39.2.309MathSciNetView Article
  13. Miura T: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Jpn. 2002, 55: 17–24.MathSciNet
  14. Miura T, Jung S-M, Takahasi S-E:Hyers-Ulam-Rassias stability of the Banach space valued differential equations y = λ y . J. Korean Math. Soc. 2004, 41: 995–1005. 10.4134/JKMS.2004.41.6.995MathSciNetView Article
  15. Miura T, Miyajima S, Takahasi S-E: A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 2003, 286: 136–146. 10.1016/S0022-247X(03)00458-XMathSciNetView Article
  16. Miura T, Miyajima S, Takahasi S-E: Hyers-Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 2003, 258: 90–96. 10.1002/mana.200310088MathSciNetView Article
  17. Cimpean DS, Popa D: On the stability of the linear differential equation of higher order with constant coefficients. Appl. Math. Comput. 2010, 217(8):4141–4146. 10.1016/j.amc.2010.09.062MathSciNetView Article
  18. Cimpean DS, Popa D: Hyers-Ulam stability of Euler’s equation. Appl. Math. Lett. 2011, 24(9):1539–1543. 10.1016/j.aml.2011.03.042MathSciNetView Article
  19. Jung S-M: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 2004, 17: 1135–1140. 10.1016/j.aml.2003.11.004MathSciNetView Article
  20. Jung S-M: Hyers-Ulam stability of linear differential equations of first order, II. Appl. Math. Lett. 2006, 19: 854–858. 10.1016/j.aml.2005.11.004MathSciNetView Article
  21. Jung S-M: Hyers-Ulam stability of linear differential equations of first order, III. J. Math. Anal. Appl. 2005, 311: 139–146. 10.1016/j.jmaa.2005.02.025MathSciNetView Article
  22. Jung S-M: Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl. 2006, 320: 549–561. 10.1016/j.jmaa.2005.07.032MathSciNetView Article
  23. Lungu N, Popa D: On the Hyers-Ulam stability of a first order partial differential equation. Carpath. J. Math. 2012, 28(1):77–82.MathSciNet
  24. Lungu N, Popa D: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 2012, 385(1):86–91. 10.1016/j.jmaa.2011.06.025MathSciNetView Article
  25. Popa D, Rasa I: The Frechet functional equation with application to the stability of certain operators. J. Approx. Theory 2012, 164(1):138–144. 10.1016/j.jat.2011.09.009MathSciNetView Article
  26. Jung S-M: Legendre’s differential equation and its Hyers-Ulam stability. Abstr. Appl. Anal. 2007., 2007: Article ID 56419. doi:10.1155/2007/56419
  27. Jung S-M: Approximation of analytic functions by Airy functions. Integral Transforms Spec. Funct. 2008, 19(12):885–891. 10.1080/10652460802321287MathSciNetView Article
  28. Jung S-M: Approximation of analytic functions by Hermite functions. Bull. Sci. Math. 2009, 133: 756–764. 10.1016/j.bulsci.2007.11.001MathSciNetView Article
  29. Jung S-M: Approximation of analytic functions by Legendre functions. Nonlinear Anal. 2009, 71(12):e103-e108. 10.1016/j.na.2008.10.007View Article
  30. Jung S-M:Hyers-Ulam stability of differential equation y + 2 x y 2 n y = 0 . J. Inequal. Appl. 2010., 2010: Article ID 793197. doi:10.1155/2010/793197
  31. Jung S-M: Approximation of analytic functions by Kummer functions. J. Inequal. Appl. 2010., 2010: Article ID 898274. doi:10.1155/2010/898274
  32. Jung S-M: Approximation of analytic functions by Laguerre functions. Appl. Math. Comput. 2011, 218(3):832–835. doi:10.1016/j.amc.2011.01.086 10.1016/j.amc.2011.01.086MathSciNetView Article
  33. Jung S-M, Rassias TM: Approximation of analytic functions by Chebyshev functions. Abstr. Appl. Anal. 2011., 2011: Article ID 432961. doi:10.1155/2011/432961
  34. Kim B, Jung S-M: Bessel’s differential equation and its Hyers-Ulam stability. J. Inequal. Appl. 2007., 2007: Article ID 21640. doi:10.1155/2007/21640
  35. Ross CC: Differential Equations - An Introduction with Mathematica. Springer, New York; 1995.
  36. Kreyszig E: Advanced Engineering Mathematics. 9th edition. Wiley, New York; 2006.

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